Solving Hamilton-Jacobi equations by minimizing residuals of monotone discretizations
This work provides a rigorous framework for solving Hamilton-Jacobi equations via residual minimization, offering theoretical guarantees and scalability to high-dimensional problems, which is important for computational PDEs and optimal control.
The authors propose solving Hamilton-Jacobi equations by minimizing the least-squares residual of monotone finite-difference discretizations, proving that critical points are unique global minimizers and deriving a posteriori error estimates. The method scales to high-dimensional problems, with condition numbers scaling as O(Δx^{-1}) for proper schemes, and numerical experiments demonstrate applicability to Eikonal, level-set, and Hamilton-Jacobi-Isaacs equations.
We establish a well-posedness and error-estimation framework that solves Hamilton-Jacobi equations by minimizing the least-squares residual of monotone finite-difference discretizations. This approach also applies naturally to second-order elliptic and parabolic problems. We prove that, under suitable monotonicity conditions, every critical point of the residual loss functional is the unique global minimizer and coincides with the solution of the discrete scheme. We derive \emph{a~posteriori} error estimates that bound the approximation error by the magnitude of the residual with explicit, computable constants, and extend the full analysis to time-dependent problems with implicit discretization of the time derivatives. A spectral analysis of the linearized system shows that the condition number scales as $O(Δx^{-1})$ for proper schemes, and as $O(\exp(Δx^{-1}))$ under a uniform ellipticity condition. These results quantify the increasing difficulty of solving the optimization problem on finer meshes, and motivates a progressive multi-level warm-start strategy using Artificial Neural Networks. Combined with the convergence theorem of Barles and Souganidis for monotone and consistent schemes, our results guarantee that the solutions obtained converge to the unique viscosity solution as the mesh is refined. Numerical experiments demonstrate the scalability of the approach to high-dimensional Eikonal equations, level-set problems, and Hamilton--Jacobi--Isaacs equations with genuine second-order diffusion arising from stochastic differential games.